What is the solution for y^{''}-2y^'+10y=x^2*e^x ?

The solution for y^{''}-2y^'+10y=x^2*e^x is y=e^x(c_{1}cos(3x)+c_{2}sin(3x))+(e^xx^2)/9-2/81 e^x

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y^{''}-2y^'+10y=x^2*e^x

y^{^{''}} - 2 y^{^{'}} + 10 y = x^{2} \cdot e^{x}

y = e^{x} (c_{1} cos (3 x) + c_{2} sin (3 x)) + \frac{e ^{x} x ^{2}}{9} - \frac{2}{81} e^{x}

Solution steps

y^{''} (x) - 2 y^{'} (x) + 10 y = x^{2} \cdot e^{x}

Solve linear ODE:

y = e^{x} (c_{1} cos (3 x) + c_{2} sin (3 x)) + \frac{e ^{x} x ^{2}}{9} - \frac{2}{81} e^{x}

y = e^{x} (c_{1} cos (3 x) + c_{2} sin (3 x)) + \frac{e ^{x} x ^{2}}{9} - \frac{2}{81} e^{x}

f (x) = (5 x - x^{2}) (5 - x - x^{2}), at x = 1

f (x) = x - 3

\int_{- 4}^{2} f (x) + 2 x + 5 d x

\frac{1}{2} cos (2 t)

\int \frac{1}{u a ^{2} - u ^{2}} d u

What is the solution for y^{''}-2y^'+10y=x^2*e^x ?
The solution for y^{''}-2y^'+10y=x^2*e^x is y=e^x(c_{1}cos(3x)+c_{2}sin(3x))+(e^xx^2)/9-2/81 e^x